Application ofMultistep InversionMethod forOnlineMonitoring Aerosol Particle Size Distribution and Aerosol Concentration

Aerosol concentration in the flow is usually time varying, and aerosol particle size distribution (PSD) is considered to be unchanged, which increases the difficulty of the measurement of aerosol PSD and concentration online. To solve these problems, a kind of multistep inversion method based on the angular light-scattering (ALS) signals is proposed. First, the aerosol PSD is estimated using shuffled frog-leaping algorithms (SFLAs) from relative ALS signals. +en, with aerosol PSD as priori information, the aerosol concentration is obtained by the Kalman filter (KF) algorithm, widely used in the real-time control system of industrial facilities for its ability of fast predictions. +e result reveals that the performance of the improved SFLA is better than that of the original SFLA in solving the aerosol PSD. Moreover, in studying the aerosol concentration, more accurate results can be obtained with larger standard deviation of process noise or smaller standard deviation of measurement noise, while decreasing sampling time interval can improve the accuracy of retrieval results and reduce time delay to a certain degree. So, to improve retrieval accuracy, the noise should be controlled, and appropriate sampling time interval should be selected. All the numerical simulations confirm that the methodology provides effective and reliable results in real-time estimating.


Introduction
Aerosols usually play a pivotal role in determining the properties of the atmosphere, e.g., by reducing visibility or affecting the net radiative fluxes and temperature [1][2][3]. Considerable researches have been carried out on studying the aerosol particle size, optical constant, and concentration in the particle dispersed medium, e.g., Aerosol Robotic Network (AERONET) and Moderate Resolution Imaging Spectroradiometer (MODIS), which are the global groundbased aerosol observation networks established to study the properties of the atmospheric aerosols [4,5]. Usually, in the atmosphere, the aerosol affects the radiative transfer, meteorological phenomena, and climate trends. e aerosol particle size distribution (PSD) and concentration are regarded as the important indicators of industrial emissions. e aerosol PSDs are usually treated as constants, while the aerosol concentration is usually changing over time, which increases the difficulty of the online monitoring of the PSDs and time-varying particle concentration [6][7][8][9]. e online particle monitoring has been widely studied [10][11][12], and the common measurement techniques for studying the properties of the particles contain optical measurement methods, acoustic measurement methods, and electric induction methods [10,13]. In the past few years, the optical measurement method (e.g., spectral extinction method, angular light-scattering method, diffraction lightscattering method, dynamic light-scattering method, laserinduced incandescence method, and laser-induced breakdown spectroscopy method) has been widely utilized to study the properties of particle system owing to offering a useful and effective approach to characterize a large number of industrial production processes. eir measurements vary from nanometer to millimeter [14]. Among these optical measurement methods, the angular light-scattering (ALS) method is regarded as one of the most viable. For example, the ALS method was applied to study the fractal dimension and size distribution of noncompact soot aggregates [15], the optical constants of the particle [16], the particle size distribution [17,18], and the particle concentration [19][20][21][22]. In the author's previous work [23,24], the ALS method combined with inverse radiative problem technique is successfully applied to study the PSD and optical constants. Moreover, the convergence accuracy and robustness of the ALS method was proved to be better than that of the spectral extinction method.
In the present study, to make the problem mathematically trackable, the aerosol particle is assumed to be spherical, and the optical properties of aerosol particles are simulated by the Mie theory. e ALS method is introduced to study the aerosol PSD and time-varying aerosol concentration. Moreover, considering the properties of the aerosol PSD and aerosol concentration mentioned above, a kind of multistep inversion method is proposed. First, the aerosol PSD is retrieved by shuffled frog-leaping algorithms (SFLAs) [25] from the relative ALS signals, which is independent of the aerosol concentration.
en, the solved aerosol PSD is regarded as known quantity, and the aerosol concentration is monitored by the Kalman filter (KF) algorithm, which performs very well in real-time control systems of industrial facilities for its ability of fast predictions, such as the particle monitoring [26,27], species monitoring in combustion applications [28,29], and others [26,28,[30][31][32]. Finally, main conclusions and prospects for further research are provided.

Angular Light-Scattering (ALS) Method.
When a collimated light with intensity I 0 impinges on a particle system, the transmitted light is scattered and absorbed by the particles. If the aerosol is optically thin and the independent scattering dominates, the real-time angular light-scattering ( Figure 1) intensity I (θ, t) at the angle θ can be expressed as follows [33][34][35][36]: where D is the diameter of the particle; D min and D max denote the lower and upper integration limits of particle size; I (θ, t) denotes the angular light-scattering intensity at angle θ and time t, which can be measured by the optical sensor; N D (t) is the unknown number concentration of suspended particle system which is a function of measurement time t; f (D) is the unknown particle size distribution (PSD) which is usually regarded as unchanged in the particle system; and i (θ, m λ , D, λ) denotes the light-scattering intensity of particles with diameter D at the angle θ from the incident direction, which can be derived in terms of the Mie scattering functions i 1 (θ, m λ , D, λ) and i 2 (θ, m λ , D, λ) [18,37,38]: where m λ , an optical constant, refers to the interaction between particles and incident light and λ denotes the incident wavelength (μm).

Shuffled Frog-Leaping Algorithms (SFLAs)
2.2.1. Original SFLA. e original SFLA, based on the evolution of memes carried by interactive individuals and global exchange of information among the interactive population, was developed by Eusuff and Lansey [39] to seek a global optimal solution by performing an informed heuristic search using a heuristic function. e concept of memetic comes from the word "meme" which can be considered as the unit of culture evolution and the sample of virtual frogs which stands for a possible solution constitutes a population.
First, the total number of the frogs S in the swarm and the number of unsolved variables N are fixed and the initial virtual frog population P X � [X 1 , X 2, . . ., X S ] is generated randomly in the original SFLA. Each frog is represented by a 1-D vector X i � [x i1 , x i2 , . . ., x iN ], which denotes a potential solution of the inverse problem. e frogs are evaluated and sorted in a descending order according to the value of fitness function, and the frog with the global best fitness is recorded and identified as X g . en, the entire swarm is divided into m memeplexes each holding n frogs such that N � m × n. e memeplexes can be regarded as the parallel frog cultures for the same goal. Each frog culture proceeds towards their goal independently.
(3) e frogs with the best and worst fitness values of each memeplex are identified as X b and X w . After the previous preparation, the evolution process is conducted to search the solution of the inverse problem. However, not all the frogs change their positions in the searching process, but only the ones with the worst fitness in each memeplex. e corresponding positions of the worst frogs are adjusted as follows: denotes the new position of the worst frog. e new generated position will be valuated after updating, and the worst frog will be replaced if the fitness function value of the new position is superior to that of the original one. Otherwise, the execution of equations (4) and (5) is repeated with respect to the global best frog. In the calculation of the step size d, the best frog in this memeplex X b is replaced by the global best frog X g . is operation is going to be repeated for E max times before taking the next step. A new random frog will be generated to replace the worst frog if there is no improvement of the new frog. All the frogs are mixed together and redivided into m memeplexes when this process is done. e information of each memeplex is passed between each other through the shuffling process. e whole process is repeated unless the iteration number reaches the user-defined iteration limit, or the fitness function value of global best position is less than the tolerance. e details of the SFLA are available in Ref. [25].

Improved SFLA.
In the SFLA, the initial positions are generated by random numbers. In this way, some of the initial positions are too close to each other. To solve these problems, the Chaos theory, the highly unstable motion of deterministic systems in finite phase space often existing in nonlinear systems, is applied to initialize the positions of the frogs. e Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect that is popularly referred to as the butterfly effect [41]. e typical logistic mapping to generate the Chaos signal is described as follows [41]: where Rand 2 is a uniformly distributed random number in the range of [0, 1] and μ is the control number; when μ � 4.0, the logistic mapping is in a fully chaotic state. Moreover, although each frog culture evolves a certain iteration without the exchange of information with other cultures to improve the local searching ability, the information exchange within each culture only proceeds between the best and worst frogs which will restrict the convergence speed and the local search ability. Each memeplex is iterated plenty of times caused by the low convergence, which occupies large amount of calculation time. To remedy these problems, the particle swarm optimization (PSO) algorithm is introduced to improve the SFLA by accelerating the convergence speed and local searching ability in each memeplexes. e basic principle of the algorithm can be explained as follows. After local search process, i.e., updating the worst frog in each memeplex, the whole frog population is shuffled and divided into new memeplexes. Subsequently, the PSO algorithm is introduced to improve the local search process in each memeplex. e velocity and position evolution equations of the worst frog in each memeplex are expressed as [42] where Rand 3 and Rand 4 are two uniformly distributed random numbers in the range of [0, 1] and C 1 and C 2 are two positive constants called acceleration coefficients. e pseudocode for the improved SFLA can be summarized as follows: (i) Step 1: input the system control parameters of the improved SFLA, i.e., the total number of frogs in the swarm S, the numbers of memeplexes m, the number of frogs in each memeplex n, the number of evolutionary iterations E max , the maximum iteration limit N c , the number of unsolved variables N, the searching space [low i , high i ] of each variable, and the tolerance for minimizing the fitness function value ε. (ii) Step 2: initialize the frogs' positions by mapping the chaotic sequence, which is generated by equation (6), to the search space. (iii) Step 3: evaluate the frogs and calculate the corresponding fitness, and then rank them in the descending order. (iv) Step 4: divide the frogs into m memeplexes according to equation (3).
Step 5: generate the new position of the worst frog in each memeplex by using equations (7) and (8). If the position of the new frog is better, then use it to update the worst frog. Otherwise, repeat this step by replacing X b with X g . (1) e iteration number reaches the user-defined iteration limit N c , iter (t) > N c (2) e fitness function value of global best position is less than the tolerance ε, Fitness (X g ) < ε

Kalman Filter Algorithm (KF).
In this study, the unknown time-varying particle number concentration in a particle dispersed system is predicted by measuring realtime ALS intensity, and the equation (1) can be transformed to a linear system and described as According to the Kalman filter algorithm, when there is white Gaussian noise added to the measurement signals, the linear system can be expressed by the following state and measurement equations [43,44]: where X(k), X(k − 1) ∈ R n denotes the n-dimensional state vector at instant k and k − 1, respectively, and X(k) � ln τ(t k ) in the present study; Y(k) ∈ R m represents the m-dimensional observation vector at instant k; Φ(k − 1) represents the state transition matrix at instant k − 1; H (k) stands for the observation matrix at instant k; and w (k − 1) and v (k) denote the mutually uncorrelated process noise at instant k − 1 and measurement noise at instant k, respectively, with zero mean and covariance matrix written as [43,44] where Q and R denote the covariance matrices of the process noise and the measurement noise, respectively; σ Q and σ R are the standard deviations of the process noise and measurement noise, respectively. To simplify the problem, the measurement noise is assumed to be additive noise and obey the standard Gaussian distribution. Usually, the Kalman filter algorithm contains two update steps. e time update mainly can be divided into state prediction and covariance prediction, while the measurement update is composed of gain calculation, state correction, and covariance correction. e time update equations and measurement update equations of the Kalman filter algorithm can be written as follows [44,45]: (i) Predicting the present stage on the basis of the previous moment stage (ii) Updating the measurement stage from the corrected prediction stage based on the present measurement signals Moreover, the posterior covariance is updated as follows: where P denotes the error covariance matrix of the state estimate vector. e flowchart of numerical procedures for retrieving the real-time particle number concentration using the KF algorithm is shown in Figure 2. First, the exact value of particle concentration is used to calculate the ALS signals, which is treated as the measurement signals in the simulation experiment without noise. en, the measurement noise and process noise are added to the measurement signals to simulate observation measurement signals with noise. Finally, the KF algorithm is applied to obtain the retrieval value of particle concentration from the obversion measurement signals with noise.

Inverse Process.
According to the theory of the inverse radiative problems [46], the inversion of multiple parameters usually requires the inverse model to have better convergence characteristics to avoid the multivalue characteristics of the retrieval results and make sure each parameter with good inverse accuracy. So, when the inverse model and algorithm are fixed, the dimension of the unsolved parameters should be reduced as much as possible, and the inversion strategies are very important.
To solve the problem, a kind of multistep inversion strategy is proposed in this study. e strategy is to perform the inversion studies on some easily accessible parameters first. en, these parameters are regarded as known quantity to reconstruct other unsolved parameters. Based on this idea, the aerosol PSDs are estimated from the relative ALS signals at first, and the mathematical expression of I R (θ) is derived as where θ 0 denotes the angle of the reference signals which can be found in Figure 1. From equation (15), it can be found that the relative ALS signals are not involved with the number concentration of suspended particle system. So, the aerosol PSDs can be reconstructed without considering the effect of the time-varying particle number concentration. Subsequently, the time-varying particle number concentration N D (t) is estimated from the real-time light-scattering intensity with the aerosol PSDs as the priori information. In this study, the aerosol PSDs are estimated by the SFLA and improved SFLA from relative ALS signals, respectively, and the time-varying particle number concentration N D (t) is estimated by the KF algorithm from the real-time lightscattering intensity. e logical relationship of numerical procedures for online monitoring the aerosol PSDs and the real-time particle number concentration is shown in Figure 3.

Numerical Simulation
According to our previous work [23], the optical constant of the aerosol particle in the present study is set as m λ � 1.53 + 0.01i, and the measurement wavelength is set as λ � 0.5 μm. All the simulations were performed on an Intel Core i7-6500 PC by using Matlab 2016. According to Ref. [46], the particle size range is set as [0.1 μm, 10 μm]. To make the problem mathematically trackable, the particle size distribution was assumed to be constant and obeys the common log-normal (L-N) and gamma function, respectively, and the corresponding mathematical expressions are expressed as where D is the average diameter of L-N distribution; σ denotes the width of the distribution; and α, β, and c are the characteristic parameters of the gamma distribution. Usually, in the modified form, c � 1, so only parameters α and β in the gamma distribution need to be investigated, respectively. e retrieval of the aerosol PSDs is solved by minimizing the fitness function value Fitness, which is defined as the sum of the square residual between the estimated and measurement signals ratios. e mathematic expression of Fitness is derived as where N θ is the number of the measurement angles. For the inverse algorithms are the stochastic optimization method and all optimizations have certain randomness, all the calculations are repeated 20 times. To ensure the online monitoring, the relative ALS signals for repeated calculations are obtained at 20 different measurement moments. e PSD is assumed to be unchanged during different measurement moments, and the average value of the 20 times retrieval results is regarded as the final retrieval results. For the purpose of investigating the reliability and feasibility of the inverse algorithms, the relative standard deviation ξ, which means the sum of the deviation between the probability distribution estimated from the inverse calculation and the true distribution of aerosol PSDs in every subinterval, is studied to evaluate the quality of inverse results, and its mathematic expression is described as Step 1 Step 2 Figure 3: Logical relationship of multistep inversion method for online monitoring the aerosol PSDs and real-time particle number concentration.

Initialization
Exact value of particle concentration N D (k)

Measuring transmittance signals Y(k)
Kalman filtering method Noise σ Q and σ R Yes No Figure 2: Flowchart of the numerical procedure for studying the real-time particle number concentration using the KF algorithm. (18) where N ′ denotes the number of subintervals which the particle size range [D min , D max ] is divided into; D i is the midpoint of the ith subinterval [D i , D i+1 ]; f X,true (D i ) is the true aerosol PSDs in the ith subinterval; and f X,est (D i ) is the estimated aerosol PSDs in the ith subinterval. e time-varying particle number concentration N D (t) in this study is assumed to obey two common time-varying distributions, e.g., sine wave-type equation (19) and lineartype equation (20), and the corresponding mathematical expressions are as follows: To investigate the reliability and feasibility of the Kalman filter algorithm, the effects of measurement noise and process noise on the estimated results are studied. e average deviation δ, which means the average value of the deviation between estimated time-varying particle number concentration N D,est (t i ) and the true time-varying particle number concentration N D,true (t i ) in each sampling time interval, is used to evaluate the estimated results, and its mathematic expression is described as where T is the total sample time and Nt denotes the number of sampling time intervals, Nt � T/Δt. Moreover, to make the problem mathematically trackable, the distribution functions of the aerosol PSDs and aerosol concentration are assumed to be known beforehand.

Retrieval of Aerosol Particle Size Distributions.
With the help of the original SFLA and improved SFLA, the aerosol PSDs are estimated from the relative ALS signals, and the real values are shown in Table 1. According to our previous work [24], the measurement angles are set as θ � 5°, 10°, 15°, and 20°, and the reference angle θ 0 � 30°. Table 1 also lists the retrieval results of aerosol PSDs using the SFLA and improved SFLA, respectively, and the corresponding inverse curves are depicted in Figure 4. It can be found that without random measurement noise, the value of relative standard deviation ξ obtained by the improved SFLA is smaller than that by the SFLA. When adding the random measurement noise to the relative ALS signals, the value of δ will increase, but it is still easy to find that the retrieval results obtained by the improved SFLA is acceptable. at is to say, the accuracy and robustness of the inverse results retrieved by the improved SFLA are higher than those by the SFLA, especially with random measurement noise. Figure 5 shows the real-time ALS signals. e black line denotes the signals without random noise, the gray line denotes the original observation signals, and the red line denotes the filtered results of the original observation signals using the KF algorithm. Figure 6 depicts the retrieval results with and without Kalman filter algorithm. e exact timedependent particle number concentration is set as the sine wave type (19), and the standard deviation of measurement noise is assumed to be 0.05, i.e., σ R � 0.05. It is obvious that if the measurement ALS signals with noise are used to reconstruct particle number concentration directly, the absolute value of maximal relative error reaches 11%. If the ALS signals are filtered by the Kalman filter algorithm beforehand, the absolute value of relative error will reduce quickly (less than 6%). e time-varying aerosol particle number concentration can be effectively reconstructed by employing the technique mentioned above.

Retrieval of Time-Varying Particle Number Concentration.
Considering the applicability of the ALS method and Kalman filter algorithm, the following research should be done to study the effects of characteristic parameters in the measurement technique on the accuracy and stability of retrieval results to choose suitable parameter settings. Figure 7 depicts estimated results obtained under different process noise.
e sampling time interval Δt is set as 0.1 s. e standard deviation of measurement noise σ R is set as 0.05. It can be found that reducing the standard deviation of process noise will result in larger deviation between real value and estimated result of aerosol particle number concentration and increase of time delay, i.e., requiring more time to response the mutation of particle number concentration. e amount of overshoot of peak value reduces with the decreasing standard deviation of process noise. ese phenomena indicate that the dynamic performance and steady-state values of the system are reduced. e main reason is that decreasing standard deviation of process noise results in the reduction of filter gain matrix K (k), an important matrix that affects the correction weight, steady-state value, and dynamic performance of the system. erefore, although the process noise and corresponding standard deviation are unknown beforehand in practical problem, a larger standard deviation of process noise should be selected to retrieve particle number concentration successfully. Figure 8 shows the effects of measurement noise on the accuracy and stability of retrieval results. e sampling time interval Δt is set as 0.1 s. e standard deviation of process noise σ Q is set as 0.01. From Figure 8, it can be found that increasing standard deviation of measurement noise will reduce the retrieval accuracy and increase time delay. e amount of overshoot of the peak value also shows increasing first and then reduces with the increasing of standard deviation of measurement noise. e reason is that according to the KF algorithm, the relationship between standard deviation of measurement noise and filtering gain K (k) is negative. Increasing the filtering gain K (k) will be conducive   Mathematical Problems in Engineering to reduce the time lag of retrieval results, and more valuable information obtained from measurement signals Y (k) rather than priori estimation vector X(k | k − 1) and last estimated results N D (t k− 1 ) will be used to improve the retrieval accuracy. erefore, although the measurement noise is unknown in practice, we should better try our best to control it to improve the retrieval accuracy.
During measurement experiments, except for these noises mentioned above affecting the measurement results, the sampling time interval Δt also plays an important role in retrieving the time-varying particle number concentration.
e standard deviations of process noise and measurement noise are set as 0.01 and 0.03, respectively. Figure 9 shows the effect of different sampling time intervals Δt on the retrieval results. It can be found that more accurate results and less time delay can be obtained, if the sampling time interval is reduced. e reason may be that more useful information about the particle system can be used to improve the retrieval accuracy when the sampling time interval Δt is smaller. Moreover, from Figure 9, it can also be found that although there are process noise and measurement noise existed in the measurement ALS signals, the retrieval results are still acceptable, if appropriate sampling time interval is selected. So, the ALS method combined with Kalman filter algorithm can be proposed as a kind of effective and reliable measurement technique to study the real-time estimation of aerosol concentration.

Conclusions
A kind of multistep inversion method based on the angular light-scattering (ALS) measurement signals is proposed to solve the online monitoring of the aerosol PSDs and aerosol concentration. e aerosol PSD is retrieved by the shuffled frog-leaping algorithms (SFLAs) from relative ALS signals, and then the time-varying aerosol concentration is obtained by the Kalman filter algorithm from the real-time ALS signals. Moreover, the influences of characteristic parameters, i.e., sampling time interval, measurement noise, and process noise, on the estimated results have been studied. e conclusions drawn from the results are as follows: (1) e convergence accuracy and robustness of the improved SFLA are higher than those of the original SFLA in studying the aerosol PSDs.
(2) With the aerosol PSDs known beforehand, the timevarying aerosol particle number concentration can be reconstructed effectively by the Kalman filter algorithm from the ALS signals even with process noise and measurement noise.
(3) e process noise and measurement noise have an important influence on the estimated results. Increasing standard deviation of process noise or reducing that of measurement noise benefit to obtaining more useful information to improve Mathematical Problems in Engineering retrieval accuracy and reduce time delay. Moreover, the reliability and stability of retrieval results are also dependent on sampling time interval, and decreasing sampling time interval can improve the accuracy of retrieval results and reduce time delay to a certain degree. So, to improve retrieval accuracy and reduce time delay, the noise should be controlled, and suitable sampling time interval should be selected.
e results obtained show that the proposed methodology is a promising approach for online monitoring of aerosol PSDs and time-varying aerosol particle number concentration. Further study will focus on performance improvement of the methodology as well as its applications in studying the nonspherical aerosol.

D:
Particle diameter (μm) d: Frog position in the shuffled frog-leaping algorithm f(D): Volume frequency distribution of particle size H: Observation matrix I: Intensity of the measurement laser (W/(m 2 ·sr)) I 0 : Intensity of incident laser (W/(m 2 ·sr)) K: Kalman gain m λ : Particle optical constant at wavelength λ M k : kth memeplexes in the shuffled frog-leaping algorithm N D : Unknown number concentration of the suspended particle system (m − 3 ·μm − 1 ) Nt: Number of sampling time intervals P: Error covariance matrix of the state estimate vector P X : Frog population in the shuffled frog-leaping algorithm Q: Process noise covariance Q ext : Extinction efficiency of particle R: Measurement noise covariance T: Total sampling time (s) t: Time (s) Δt: Sampling time interval (s) v: Measurement noise matrix V i : Velocity of the particle in the PSO algorithm w: Process noise matrix X: State vector X S : Sth frog in the shuffled frog-leaping algorithm Greek symbols Φ: State transition matrix λ: Incident wavelength of the laser (μm) μ: Control number in the Chaos theory σ: Sum of the deviation between retrieval result and real value for N D (t) ξ: Sum of the deviation between retrieval result and real value for f (D) σ Q : Standard deviation of process noise σ R : Standard deviation of measurement noise Subscripts est: Estimated value L-N: Log-normal distribution Gamma: Gamma distribution true: True value.

Data Availability
Some or all data, models, or code generated or used during the study are available from the corresponding author by request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.